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If $B$ and $B'$ are the matrix representations of a bilinear form in two bases, then these matrices are related by the equation $T^t B T = B'$ for an invertible matrix $T$.

Is it the case that automatically $T^t = T^{- 1}$ for bilinear forms, which yields the well-known transformation of matrix representations of endomorphisms under a change in basis, or does the basis transformation for bilinear forms only work for $T^t B T = B'$ and not necessarily for $T^{-1}BT=B'$?

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I tried to fix the English according to what I think you are asking. –  KCd Jul 8 '12 at 14:41
It is definitely not the case that $T^t = T^{-1}$ when $T$ is a change of basis matrix in general. The equations relating different matrix representations of a bilinear form and different matrix representations of an endomorphism are not the same equations. –  KCd Jul 8 '12 at 14:42

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